Heat conduction equation in the physics of asteroids and meteoroids
Leoš Pohl
MFF UK, AUUK, V Holešovičkách 2, 180 00 Praha 8
Introduction
The influence of thermal emission from the surfaces of asteroids and meteoroids
on their orbital and rotational motion is a frequently discussed application of the Heat
Conduction Equation (HCE). Because its analytical solution is difficult (if not
impossible), we developed a numerical solver in Fortran that employs Finite Difference
Methods (FDMs). The program calculates temperature distribution within a system of
1-dimensional slabs which approximate an asteroid. We conducted various tests of the
FDMs aimed at their accuracy, convergence and computational efficiency. We also
discussed the influence of grid spacing. We attempted a simplified analytical solution of
the HCE which was compared with the numerical results. The force resulting from the
departure of thermal photons was evaluated together with the drift of semimajor axis of
sample asteroids. We compared the results with values from literature. The drift is
called the Yarkovsky effect and we discussed its role in the transport of meteorites to
the near-Earth space and in the evolution of asteroid families. We also showed that the
thermal emission of photons can cause torques that affect the spin of asteroids. We
illustrated this so-called YORP effect on an alignment of spin axes within the Slivan
Group.
Applications of the HCE in the Physics of the Solar System
The Yarkovsky effect
The Yarkovsky effect is an orbital perturbation
resulting from infrared emission and thermal inertia of an asteroid. It is usually divided
into two components – the diurnal effect and the seasonal effect. Let’s assume that the
asteroid’s spin axis is perpendicular to the orbital plane. As the asteroid rotates about its
axis being insolated by the Sun, the sun-facing side heats up causing radiation of
infrared photons and subsequent loss of asteroid's momentum. The Boltzmann law
implies that the loss of momentum is higher in areas with higher temperature and lower
in the areas with lower temperature and thus a recoil force may act on the asteroid body.
The thermal inertia of the asteroid causes the temperature to take the maximum value
later than the insolation function and so does the loss of momentum. This gives rise to
acceleration along the orbital motion of the asteroid which can lead to substantial
semimajor axis drift. The semimajor axis drift can be estimated from a simplified Gauss
equation –
, where
is the asteroid’s mean motion and
the transversal
component of acceleration. The seasonal effect is related to the asteroid’s orbital motion
and it always decays the orbit of the asteroid. The Yarkovsky effect depends on the size
of the asteroid
. The diurnal effect is the largest for centimeter to meter
sized bodies, the seasonal effect for about 10 meter bodies.
We discussed the Yarkovsky effect, and its role in the transport of meteorites to
the Earth and in the evolution of asteroid families.
The YORP effect
Let's have an asteroid of a perfectly spherical shape. Let's
attach a wedge to the asteroid's equator. The photons departing from the insolated
wedge face carry away momentum in direction that lies in the equatorial plane which
either spins the asteroid up or down. The momentum the photons carry away in the
direction perpendicular to the plane changes the obliquity. The YORP effect is
illustrated on a preferential alignment of spin axes within the Slivan Group.
Mathematical and Physical Background
We have an asteroid orbiting the Sun at an angular speed
axis at
and rotating about its
. As the asteroid absorbs heat flux from the Sun, the temperature of its sun-
facing side rises. If we were standing on the equator and measuring the temperature on
the surface and the rotation axis was in the orbiting plane and perpendicular to the Sun
direction vector (originating in the centre of the asteroid and aiming at the Sun), we
would find out that the temperature would be the highest in the afternoon rather than at
noon. This is due to a property of materials which can be called thermal inertia. Taking
into account the Stefan-Boltzmann law,
Boltzmann constant,
- radiation area,
(
- emissivity,
- Stefan-
the temperature on ) , the photon radiation is
the highest in the afternoon too. However, the loss of photons implies additional nongravitational forces that act on the asteroid.
To evaluate these forces, we need to know the temperature distribution on the
surface and thus solve the HCE. We solve only one-dimensional HCE. Nonetheless, the
1-D simplification yields useful results. We can divide the asteroid into thin 1-D slabs
originating in the centre and extending up to an surface point. In one dimension the
HCE takes the form:
where
is the conductivity of the asteroid’s material in
density in
and
,
,
is the bulk
is the specific heat capacity of the material in
,
denote derivatives of temperature w.r.t. time and space variables and
is
an equilibrium temperature1 that is determined from the heat radiation incident on the
object. The
is the temperature on the surface and it is determined from energy
conservation on the surface (
):
We use FDMs to numerically approximate solution of the HCE. This is achieved
by transforming the above HCE into a finite difference equation. We use 3 FDMs. They
differ according to what kind of difference is used to replace the time derivative: 1) we
replace the time derivative with a forward difference (FTCS), 2) a backward difference
(BTCS) and 3) a central difference (CTCS). In all cases the space derivate is replaced by
the central difference. Unlike the FTCS method, the BTCS and CTCS methods require
solving a set of linear equations, however, our boundary condition is a fourth-order
polynomial and supplying the condition to the set of equations transforms it to a nonlinear one. To avoid this non-linear set, we have to approximate the subsurface
temperature (
) which appears in the boundary condition after the
transformation to finite differences). We implemented 3 methods to approximate the
subsurface
temperature
–
the
BTCS1
method
replaces
with
(i.e. the subsurface temperature from the previous time step); the
BTCS2 method starts also with the subsurface temperature from the previous time step,
it then solves the set of equation leading to
which is then used in the
boundary condition and so forth until a required precision is achieved. The BTCS3
method calculates
using a FTCS step. Although the FTCS method does not
solve a set of equations which makes it a very efficient algorithm, however, it can be
1
The fact that we assume the temperature
in the centre of the asteroid remains unchanged, i.e. the
temperature variations on the surface do not propagate to the centre.
shown (Press et al. 1987) that the FTCS method only converges for
(so
called von Neumann stability criterion) limiting the usable grids.
Comparison of the methods and analytical solution
We showed that the methods successfully converge as the grids become denser. In
terms of accuracy we consider the BTCS2 method the most accurate.
Despite several simplifications and an assumption of a harmonic insolation function, we
can conclude that the derived analytical solution provides some good first-guess
estimates.
We also estimated force on 4 sample asteroids of the order
. The
transversal accelerations (along the direction of orbital motion) of the asteroids ranged
between
to
contributes
to
. Based on the transversal acceleration which
the
semimajor
for a
bulk density
drift
( )
most
significantly,
we
estimated
basalt asteroid (heat conductivity
, specific heat capacity
) with
period. For a regolith asteroid (conductivity
), we estimated
,
rotation
, bulk density
. These values are in order-of-
magnitude agreement with values from literature.
Acknowledgements
I
would
like
to
express
my
sincere
gratitude
to
my
supervisor
Mgr. Miroslavov Brož Ph.D. for his friendly approach, for the time he spent discussing
the topics and evaluating my work and for valuable insights.
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